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Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremalimdvK 27801* Add universal quantifier to both sides of an equivalence. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Part of Lemma 5 of [KalishMontague] p. 86. (The other parts are just notbii 289 and imbi12i 318.) (Contributed by NM, 11-Apr-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
TheoremalbidK 27802 Add universal quantifier to both sides of an equivalence. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 19-Apr-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
TheoremalbidvK 27803* Add universal quantifier to both sides of an equivalence. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 11-Apr-2017.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theorema4imK 27804* Specialization. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Lemma 8 of [KalishMontague] p. 87. (Contributed by NM, 19-Apr-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theorema4imvK 27805* Specialization. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Lemma 8 of [KalishMontague] p. 87. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremax4wfK 27806* Weak version of ax-4 1692. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Lemma 9 of [KalishMontague] p. 86. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. x ph 
 ->  A. y A. x ph )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremax4wK 27807* Weak version of ax-4 1692. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Lemma 9 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremax4vK 27808* Version of ax-4 1692 when  x does not occur in  ph. This provides the other direction of ax-17 1628. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 10-Apr-2017.)
 |-  ( A. x ph  ->  ph )
 
Theorem19.8vK 27809* Version of 19.8a 1758 and its converse when  x does not occur in  ph. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 10-Apr-2017.)
 |-  ( ph 
 <-> 
 E. x ph )
 
TheoremcbvaliK 27810* Change bound variable. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalivK 27811* Change bound variable. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalK 27812* Change bound variable. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 9-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps  ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvalvK 27813* Change bound variable. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvexvK 27814* Change bound variable. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremax6wfK 27815* Weak version of ax-6 1534 from which we can prove any ax-6 1534 instance not involving wff variables or bundling. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 19-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps  ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( -.  A. y ps  ->  A. x  -.  A. y ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremax6wK 27816* Weak version of ax-6 1534 from which we can prove any ax-6 1534 instance not involving wff variables or bundling. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremhba1wK 27817* Weak version of hba1 1718. See comments for ax6wK 27816. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  A. x A. x ph )
 
Theoremhbe1wK 27818* Weak version of hbe1 1565. See comments for ax6wK 27816. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph 
 ->  A. x E. x ph )
 
Theoremax7wK 27819* Weak version of ax-7 1535 from which we can prove any ax-7 1535 instance not involving wff variables or bundling. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. Unlike ax-7 1535, this theorem requires that  x and  y be distinct i.e. are not bundled. See the description in the comment of equidK 27792. (Contributed by NM, 10-Apr-2017.)
 |-  (
 y  =  z  ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
TheoremhbalwK 27820* Weak version of hbal 1567. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. Unlike hbal 1567, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  z  ->  (
 ph 
 <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremax7dgenK 27821 Degenerate instance of ax-7 1535 where bundled variables  x and  y have a common substitution. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 13-Apr-2017.)
 |-  ( A. x A. x ph  ->  A. x A. x ph )
 
Theoremax11wflemK 27822 Verson of ax11wlemK 27823 without distinct variables. (Contributed by NM, 19-Apr-2017.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax11wlemK 27823* Lemma for weak version of ax-11 1624. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. In some cases, this lemma may lead to shorter proofs than ax11wK 27824. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax11wK 27824* Weak version of ax-11 1624 from which we can prove any ax-7 1535 instance not involving wff variables or bundling. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. An instance of the first hypothesis will normally require that  x and  y be distinct (unless  x does not occur in  ph). See the description in the comment of equidK 27792. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   =>    |-  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax11dgenK 27825 Degenerate instance of ax-11 1624 where bundled variables  x and  y have a common substitution. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 13-Apr-2017.)
 |-  ( x  =  x  ->  (
 A. x ph  ->  A. x ( x  =  x  ->  ph ) ) )
 
Theoremax11wdemoK 27826* Example of an application of ax11wK 27824 that results in an instance of ax-11 1624 for a contrived formula with mixed free and bound variables,  ( x  e.  y  /\  A. x
z  e.  x  /\  A. y A. z y  e.  x ), in place of  ph. The proof illustrates bound variable renaming with cbvalvK 27813 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 14-Apr-2017.)
 |-  ( x  =  y  ->  (
 A. y ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )  ->  A. x ( x  =  y  ->  ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )
 ) ) )
 
Theoremax12wK 27827* Weak version (principal instance) of ax-12 1633 not involving bundling. Uses only Tarski's predicate calculus axiom schemes. The proof is trivial but is included to complete the set ax6wK 27816, ax7wK 27819, and ax11wK 27824. See the description in the comment of equidK 27792. (Contributed by NM, 10-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )
 
Theoremax12dgen1K 27828 Degenerate instance of ax-12 1633 where bundled variables  x and  y have a common substitution. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  z 
 ->  A. x  x  =  z ) )
 
Theoremax12dgen2K 27829 Degenerate instance of ax-12 1633 where bundled variables  x and  z have a common substitution. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  x 
 ->  A. x  y  =  x ) )
 
Theoremax12dgen3K 27830 Degenerate instance of ax-12 1633 where bundled variables  y and  z have a common substitution. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  y 
 ->  A. x  y  =  y ) )
 
Theoremax12dgen4K 27831 Degenerate instance of ax-12 1633 where bundled variables  x,  y, and  z have a common substitution. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. See the description in the comment of equidK 27792. (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  x 
 ->  A. x  x  =  x ) )
 
16.22.2  Derive ax-12o from ax-12
 
Theoremax12vX 27832* A weaker version of ax-12 1633 with distinct variable restrictions on pairs  x ,  z and  y ,  z. In order to show that this weakening is adequate, this should be the only theorem referencing ax-12 1633 directly. (Contributed by NM, 30-Jun-2016.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )
 
Theoremax12o10lem1X 27833 Lemma for ax12o 1663 and ax10 1677. Same as equcomi 1822, using only Tarski's FOL axiom schemes (see description for equidK 27792).

Note that in these lemmas we use ax-9v 1632 instead of ax-9 1684 since the proof of ax9 1683 from ax-9v 1632 makes use of ax-12o 1664. The first use of ax-12o 1664 occurs in ax10lem24 1673. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

 |-  ( x  =  y  ->  y  =  x )
 
Theoremax12o10lem2X 27834 Lemma for ax12o 1663 and ax10 1677. Same as equequ1 1829, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( x  =  y  ->  ( x  =  z  <->  y  =  z
 ) )
 
Theoremax12o10lem3X 27835 Lemma for ax12o 1663 and ax10 1677. Same as ax4 1691, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremax12o10lem3K 27836* Lemma for ax12o 1663 and ax10 1677. Weak version of ax4 1691, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremax12o10lem4K 27837* Lemma for ax12o 1663. Weak version of 19.8a 1758, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ph  ->  E. x ph )
 
Theoremax12o10lem4X 27838 Lemma for ax12o 1663. Same as 19.8a 1758, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 7-Nov-2015.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ph )
 
Theoremax12o10lem5K 27839* Lemma for ax12o 1663 and ax10 1677. Weak version of hba1 1718, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  A. x A. x ph )
 
Theoremax12o10lem5X 27840 Lemma for ax12o 1663 and ax10 1677. Same as hba1 1718, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( A. x ph  ->  A. x A. x ph )
 
Theoremax12o10lem6K 27841* Lemma for ax12o 1663 and ax10 1677. Weak version of hbn 1722, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
Theoremax12o10lem6X 27842 Lemma for ax12o 1663 and ax10 1677. Same as hbn 1722, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( -.  ph  ->  A. x  -.  ph )
 
Theoremax12o10lem7K 27843* Lemma for ax12o 1663 and ax10 1677. Weak version of hbimd 1809, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  A. x ( ps 
 ->  ch ) ) )
 
Theoremax12o10lem7X 27844 Lemma for ax12o 1663 and ax10 1677. Same as hbimd 1809, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  A. x ( ps 
 ->  ch ) ) )
 
Theoremax12o10lem8K 27845* Lemma for ax12o 1663 and ax10 1677. Weak version of a4ime 1868 with distinct variables, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  w  ->  ( ps  <->  ta ) )   &    |-  ( x  =  z  ->  (
 ph  ->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ps )
 
Theoremax12o10lem8X 27846* Lemma for ax12o 1663 and ax10 1677. Similar to a4ime 1868 with distinct variables, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( x  =  z  ->  (
 ph  ->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( ph  ->  E. x ps )
 
Theoremax12o10lem9K 27847* Lemma for ax12o 1663. Weak version of ax6o 1696, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  ( -.  A. x  -.  A. x ph  -> 
 ph )
 
Theoremax12o10lem10K 27848* Lemma for ax12o 1663. Weak version of hbnt 1717, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( ph  ->  A. x ph )  ->  ( -.  ph  ->  A. x  -.  ph ) )
 
Theoremax12o10lem9X 27849 Lemma for ax12o 1663. Same as ax6o 1696 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax12o10lem10X 27850 Lemma for ax12o 1663. Same as hbnt 1717 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( A. x ( ph  ->  A. x ph )  ->  ( -.  ph  ->  A. x  -.  ph ) )
 
Theoremax12o10lem11K 27851* Lemma for ax12o 1663. Weak version of hbim 1723, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ch ) )   &    |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  (
 ( ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
 
Theoremax12o10lem11X 27852 Lemma for ax12o 1663. Same as hbim 1723 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  (
 ( ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
 
Theoremax12o10lem12K 27853* Lemma for ax12o 1663. Weak version of 19.9t 1761, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( ph  ->  A. x ph )  ->  ( E. x ph 
 ->  ph ) )
 
Theoremax12o10lem12X 27854 Lemma for ax12o 1663. Same as 19.9t 1761 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( A. x ( ph  ->  A. x ph )  ->  ( E. x ph  ->  ph ) )
 
Theoremax12o10lem13K 27855* Lemma for ax12o 1663. Weak version of 19.9 1762, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( E. x ph  <->  ph )
 
Theoremax12o10lem13X 27856 Lemma for ax12o 1663. Same as 19.9 1762 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. x ph  <->  ph )
 
Theoremax12o10lem14K 27857* Lemma for ax12o 1663. Weak version of 19.23 1777, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  w  ->  ( ps  <->  ta ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) )
 
Theoremax12o10lem14X 27858 Lemma for ax12o 1663. Same as 19.23 1777 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   =>    |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) )
 
Theoremax12o10lem15K 27859* Lemma for ax12o 1663. Weak version of exlimih 1782, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  w  ->  ( ps  <->  ta ) )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremax12o10lem15X 27860 Lemma for ax12o 1663. Same as exlimih 1782 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
TheoremequvinvK 27861* Similar to equvini 1879. Uses only Tarski's FOL axiom schemes (see description for equidK 27792). Note that  x and  y are bundled. (Contributed by NM, 20-Apr-2017.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremax12olem16K 27862* Lemma for ax12o 1663. Weak version of equsalh 1851, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  ( ps  <->  ta ) )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
Theoremax12olem16X 27863* Lemma for ax12o 1663. Weaker version of equsalh 1851 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
Theoremax12olem17K 27864* Lemma for ax12o 1663. Weak version of 19.21 1771, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  w  ->  ( ps  <->  ta ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremax12olem17X 27865 Lemma for ax12o 1663. Same as 19.21 1771 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremax12olem18K 27866* Lemma for ax12o 1663. Weak version of hbim1 1810, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  w  ->  ( ps  <->  ta ) )   &    |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ( ph  ->  ps )  ->  A. x (
 ph  ->  ps ) )
 
Theoremax12olem18X 27867 Lemma for ax12o 1663. Same as hbim1 1810 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ( ph  ->  ps )  ->  A. x (
 ph  ->  ps ) )
 
Theoremax12olem19K 27868* Lemma for ax12o 1663. Weak version of nfex 1733, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( E. y ph  ->  A. x E. y ph )
 
Theoremax12olem19X 27869 Lemma for ax12o 1663. Same as nfex 1733 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 20-Dec-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. y ph  ->  A. x E. y ph )
 
Theoremax12olem20K 27870* Lemma for ax12o 1663. Weak version of 19.12 1766, using only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  (
 y  =  w  ->  ( ph  <->  ch ) )   =>    |-  ( E. x A. y ph  ->  A. y E. x ph )
 
Theoremax12olem20X 27871 Lemma for ax12o 1663. Same as 19.12 1766 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 20-Dec-2015.) (New usage is discouraged.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
Theoremax12olem21K 27872* Lemma for ax12o 1663. Similar to equvin 1999 but with a negated equality. Uses only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  ( E. w ( y  =  w  /\  -.  z  =  w )  <->  -.  y  =  z )
 
Theoremax12olem21X 27873* Lemma for ax12o 1663. Similar to equvin 1999 but with a negated equality. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  ( E. w ( y  =  w  /\  -.  z  =  w )  <->  -.  y  =  z )
 
TheoremequextvK 27874* An extensionality-like property for equality. Similar to equveli 1880. Uses only Tarski's FOL axiom schemes (see description for equidK 27792). Note that  x and  y are bundled. (Contributed by NM, 20-Apr-2017.)
 |-  ( x  =  y  <->  A. z ( z  =  x  <->  z  =  y
 ) )
 
Theoremax12olem22K 27875* Lemma for ax12o 1663. Negate the equalities in ax-12 1633, shown as the hypothesis. Uses only Tarski's FOL axiom schemes (see description for equidK 27792). Note that  x and  y are bundled. (Contributed by NM, 20-Apr-2017.) (New usage is discouraged.)
 |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -.  y  =  z  ->  A. x  -.  y  =  z ) )
 
Theoremax12olem22X 27876* Lemma for ax12o 1663. Negate the equalities in ax-12 1633, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -.  y  =  z  ->  A. x  -.  y  =  z ) )
 
Theoremax12olem23aK 27877 Lemma for ax12o 1663. Show one direction of the equivalence of an intermediate equivalent to ax-12o 1664 with the conjunction of ax-12 1633 and a variant with negated equalities. Uses only Tarski's FOL axiom schemes (see description for equidK 27792). (Contributed by NM, 19-Apr-2017.) (New usage is discouraged.)
 |-  (
 ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )  /\  ( -.  x  =  y  ->  ( -.  y  =  z 
 ->  A. x  -.  y  =  z ) ) ) 
 ->  ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) ) )
 
Theoremax12olem23X 27878 Lemma for ax12o 1663. Show the equivalence of an intermediate equivalent to ax-12o 1664 with the conjunction of ax-12 1633 and a variant with negated equalities. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  (
 ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )  <-> 
 ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )  /\  ( -.  x  =  y  ->  ( -.  y  =  z 
 ->  A. x  -.  y  =  z ) ) ) )
 
Theoremax12olem24K 27879* Lemma for ax12o 1663. Construct an intermediate equivalent to ax-12 1633 from two instances of ax-12 1633. Uses only Tarski's FOL axiom schemes (see description for equidK 27792). Note that  x and  y are bundled. THIS IS AS FAR AS I WAS ABLE TO GET using only Tarski's schemes; I couldn't find a way to prove ax12olem25 1659 through ax12olem27 1661 without making  x and  y distinct. (Contributed by NM, 21-Apr-2017.) (New usage is discouraged.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )   &    |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -. 
 A. x  -.  y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12olem24X 27880* Lemma for ax12o 1663. Construct an intermediate equivalent to ax-12 1633 from two instances of ax-12 1633. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )   &    |-  ( -.  x  =  y  ->  ( y  =  w 
 ->  A. x  y  =  w ) )   =>    |-  ( -.  x  =  y  ->  ( -. 
 A. x  -.  y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12olem25X 27881 Lemma for ax12o 1663. See ax12olem27 1661 for derivation of ax-12o 1664 from the conclusion. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )
 
Theoremax12olem26X 27882* Lemma for ax12o 1663. Same as dvelimfALT2 27926 without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   &    |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremax12olem27X 27883* Lemma for ax12o 1663. Derivation of ax-12o 1664 from the hypotheses, without using ax-12o 1664. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  z  ->  ( z  =  w  ->  A. x  z  =  w )
 )   &    |-  ( -.  A. x  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
 
Theoremax12olem28X 27884* Lemma for ax12o 1663. Derivation of ax-12o 1664 from the hypotheses, without using ax-12o 1664. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.)
 |-  ( -.  x  =  z  ->  ( -.  A. x  -.  z  =  w  ->  A. x  z  =  w ) )   &    |-  ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  w  ->  A. x  y  =  w ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
 
Theoremax12oX 27885 Derive set.mm's original ax-12o 1664 from the shorter ax-12 1633.

Our current practice is to use axiom ax-12o 1664 from here on (except in the proofs of ax10 1677 and ax9 1683 below) instead of theorem ax12o 1663 in order to standardize the use ax-9 1684 instead of ax-9v 1632. Note that the derivation of ax-9 1684 from ax-9v 1632 (theorem ax9 1683 below) makes use of ax12o 1663; thus we use ax-9v 1632 to prove ax12o 1663 to avoid a circular argument . (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.)

 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y 
 ->  ( x  =  y 
 ->  A. z  x  =  y ) ) )
 
16.22.3  Derive ax-10
 
Theoremax10lem16X 27886 Lemma for ax10 1677. Similar to equequ2 1830, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( x  =  y  ->  ( z  =  x  <->  z  =  y
 ) )
 
Theoremax10lem17X 27887 Lemma for ax10 1677. Similar to hban 1724, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  (
 ( ph  /\  ps )  ->  A. x ( ph  /\ 
 ps ) )
 
Theoremax10lem18X 27888 Lemma for ax10 1677. Similar to exlimih 1782, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremax10lem19X 27889* Lemma for ax10 1677. Similar to cbv3ALT 1875 with distinct variables, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremax10lem20X 27890* Change bound variable without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 22-Jul-2015.) (New usage is discouraged.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
Theoremax10lem21X 27891* Change free variable without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. x  x  =  z )
 
Theoremax10lem22X 27892* Lemma for ax10 1677. Similar to ax-10 1678 but with distinct variables, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremax10lem23X 27893 Lemma for ax10 1677. Similar to ax-10o 1835 but with reversed antecedent, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( A. y  y  =  x  ->  ( A. x ph 
 ->  A. y ph )
 )
 
Theoremax10lem24X 27894* Lemma for ax10 1677. Similar to dvelim 2092 with first hypothesis replaced by distinct variable condition, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremax10lem25X 27895* Lemma for ax10 1677. Similar to dveeq2 1928, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 20-Jul-2015.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
 )
 
Theoremax10lem26X 27896* Distinctor with bound variable change without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 8-Jul-2016.) (New usage is discouraged.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  x )
 
Theoremax10lem27X 27897* Change free and bound variables without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 22-Jul-2015.) (New usage is discouraged.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremax10X 27898 Proof of axiom ax-10 1678 from others, without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (See remarks for ax12o10lem1 1635 about why we use ax-9v 1632 instead of ax-9 1684.)

Our current practice is to use axiom ax-10 1678 from here on instead of theorem ax10 1677, in order to preferentially use ax-9 1684 instead of ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (New usage is discouraged.)

 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theorema16gALTX 27899* Alternate proof of a16g 2000 without using ax-4 1692, ax-9 1684, or ax-10 1678 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
 
TheoremalequcomX 27900 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
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